Hardware implementations of a quasi-cyclic syndrome decoder

ABSTRACT

Disclosed are devices, systems and methods for providing hardware implementations of a quasi-cyclic syndrome decoder. An example method of reducing the complexity of a decoder includes receiving a noisy codeword that is a based on a transmitted codeword generated from a quasi-cyclic linear code; computing a plurality of syndromes based on the noisy codeword; selecting a first syndrome from the plurality of syndromes; generating a memory cell address as a function of the first syndrome; reading, based on the memory cell address, a coset leader corresponding to the first syndrome; and determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.

TECHNICAL FIELD

This patent document generally relates to non-volatile memory devices, and more specifically, to error correction in non-volatile memory devices.

BACKGROUND

Data integrity is an important feature for any data storage device and data transmission. Use of strong error-correction codes (ECCs) is recommended for various types of data storage devices including NAND flash memory devices.

Solid-state drives (SSDs) use multi-level NAND flash devices for persistent storage. However, the multi-level NAND flash devices can be inherently unreliable and generally need to use ECCs to allow dramatic increase in data reliability at the expense of extra storage space for ECC parity bits. There is a demand for ECCs that can provide robust and reliable data protection with low-complexity hardware implementations.

SUMMARY

Embodiments of the disclosed technology relate to methods, devices and systems for providing low-complexity hardware implementations of a quasi-cyclic syndrome decoder.

In an example aspect, a method for reducing the complexity of a decoder includes receiving a noisy codeword that is a based on a transmitted codeword generated from a quasi-cyclic linear code; computing a plurality of syndromes based on the noisy codeword; selecting a first syndrome from the plurality of syndromes; generating a memory cell address as a function of the first syndrome; reading, based on the memory cell address, a coset leader corresponding to the first syndrome; and determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.

In another example aspect, the above-described method may be implemented by a video encoder apparatus or a video decoder apparatus that comprises a processor.

In yet another example aspect, these methods may be embodied in the form of processor-executable instructions and stored on a computer-readable program medium.

The subject matter described in this patent document can be implemented in specific ways that provide one or more of the following features.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an example of a memory system.

FIG. 2 is an illustration of an example non-volatile memory device.

FIG. 3 is an example diagram illustrating the cell voltage level distribution (Vth) of a non-volatile memory device.

FIG. 4 is another example diagram illustrating the cell voltage level distribution (Vth) of a non-volatile memory device.

FIG. 5 is an example diagram illustrating the cell voltage level distribution (Vth) of a non-volatile memory device before and after program interference.

FIG. 6 is an example diagram illustrating the cell voltage level distribution (Vth) of a non-volatile memory device as a function of the reference voltage.

FIG. 7 is a block diagram of an example quasi-cyclic code syndrome-based decoder.

FIG. 8 illustrates a flowchart of another example method for reducing the latency of a quasi-cyclic linear code decoder.

DETAILED DESCRIPTION

Flash memory is ubiquitous in portable electronic devices, such as computers, digital cameras, digital music players, cellular telephones, personal data assistants (PDAs), or the like. These host devices communicate with the flash memory using a flash translation layer (FTL), which is a table is used to divert the address of any host request (called a logical block address (LBA)) to the real location that the corresponding data was stored in the flash memory (called a physical block address (PBA)). The size of this table is typically quite large, and is usually stored in dynamic random access memory (DRAM).

Typically, the FTL performs at least the following functions:

-   -   Write updated information to a new empty page and divert all the         subsequent requests to its new address     -   Maintain wear-leveling by uniformly using the available pages     -   Efficiently perform a garbage collection operation for all the         used blocks

FIGS. 1-6 overview a non-volatile memory system in which embodiments of the disclosed technology may be implemented.

FIG. 1 is a block diagram of an example of a memory system 100 implemented based on some embodiments of the disclosed technology. The memory system 100 includes a memory module 110 that can be used to store information for use by other electronic devices or systems. The memory system 100 can be incorporated (e.g., located on a circuit board) in other electronic devices and systems. Alternatively, the memory system 100 can be implemented as an external storage device such as a USB flash drive and a solid-state drive (SSD).

The memory module 110 included in the memory system 100 can include memory areas (e.g., memory arrays) 102, 104, 106, and 108. Each of the memory areas 102, 104, 106, and 108 can be included in a single memory die or in multiple memory dice. The memory die can be included in an integrated circuit (IC) chip.

Each of the memory areas 102, 104, 106, and 108 includes a plurality of memory cells. Read, program, or erase operations can be performed on a memory unit basis. Thus, each memory unit can include a predetermined number of memory cells. The memory cells in a memory area 102, 104, 106, or 108 can be included in a single memory die or in multiple memory dice.

The memory cells in each of memory areas 102, 104, 106, and 108 can be arranged in rows and columns in the memory units. Each of the memory units can be a physical unit. For example, a group of a plurality of memory cells can form a memory unit. Each of the memory units can also be a logical unit. For example, the memory unit can be a bank, block, or page that can be identified by a unique address such as bank address, block address, and page basis address. During a read or write operation, the unique address associated with a particular memory unit can be used to access that particular memory unit. Based on the unique address, information can be written to or retrieved from one or more memory cells in that particular memory unit.

The memory cells in the memory areas 102, 104, 106, and 108 can include non-volatile memory cells. Examples of non-volatile memory cells include flash memory cells, phase change memory (PRAM) cells, magnetoresistive random-access memory (MRAM) cells, or other types of non-volatile memory cells. In an example implementation where the memory cells are configured as NAND flash memory cells, the read or write operation can be performed on a page basis. However, an erase operation in a NAND flash memory is performed on a block basis.

Each of the non-volatile memory cells can be configured as a single-level cell (SLC) or multiple-level memory cell. A single-level cell can store one bit of information per cell. A multiple-level memory cell can store more than one bit of information per cell. For example, each of the memory cells in the memory areas 102, 104, 106, and 108 can be configured as a multi-level cell (MLC) to store two bits of information per cell, a triple-level cell (TLC) to store three bits of information per cell, or a quad-level cells (QLC) to store four bits of information per cell. In another example, each of the memory cells in memory area 111 can be configured to store at least one bit of information (e.g., one bit of information or multiple bits of information), and each of the memory cells in memory area 112 can be configured to store more than one bit of information.

As shown in FIG. 1, the memory system 100 includes a controller module 120. The controller module 120 includes a memory interface 121 to communicate with the memory module 110, a host interface 126 with communicate with a host (not shown), a processor 124 to executes firmware-level code, and caches and memories 122 and 123 to temporarily or persistently store executable firmware/instructions and associated information. In some implementations, the controller unit 120 can include an error correction engine 125 to perform error correction operation on information stored in the memory module 110. Error correction engine 122 can be configured to detect/correct single bit error or multiple bit errors. In another implementation, error correction engine 125 can be located in the memory module 110.

The host can be a device or a system that includes one or more processors that operate to retrieve data from the memory system 100 or store or write data into the memory system 100. In some implementations, examples of the host can include a personal computer (PC), a portable digital device, a digital camera, a digital multimedia player, a television, and a wireless communication device.

In some implementations, the controller module 120 can also include a host interface 126 to communicate with the host. Host interface 126 can include components that comply with at least one of host interface specifications, including but not limited to, Serial Advanced Technology Attachment (SATA), Serial Attached Small Computer System Interface (SAS) specification, Peripheral Component Interconnect Express (PCIe).

FIG. 2 illustrates an example of a memory cell array implemented based on some embodiments of the disclosed technology.

In some implementations, the memory cell array can include NAND flash memory array that is partitioned into many blocks, and each block contains a certain number of pages. Each block includes a plurality of memory cell strings, and each memory cell string includes a plurality of memory cells.

In some implementations where the memory cell array is NAND flash memory array, read and write (program) operations are performed on a page basis, and erase operations are performed on a block basis. All the memory cells within the same block must be erased at the same time before performing a program operation on any page included in the block. In an implementation, NAND flash memories may use an even/odd bit-line structure. In another implementation, NAND flash memories may use an all-bit-line structure. In the even/odd bit-line structure, even and odd bit-lines are interleaved along each word-line and are alternatively accessed so that each pair of even and odd bit-lines can share peripheral circuits such as page buffers. In all-bit-line structure, all the bit-lines are accessed at the same time.

FIG. 3 illustrates an example of threshold voltage distribution curves in a multi-level cell device, wherein the number of cells for each program/erase state is plotted as a function of the threshold voltage. As illustrated therein, the threshold voltage distribution curves include the erase state (denoted “ER” and corresponding to “11”) with the lowest threshold voltage, and three program states (denoted “P1”, “P2” and “P3” corresponding to “01”, “00” and “10”, respectively) with read voltages in between the states (denoted by the dotted lines). In some embodiments, each of the threshold voltage distributions of program/erase states has a finite width because of differences in material properties across the memory array.

In writing more than one data bit in a memory cell, fine placement of the threshold voltage levels of memory cells is needed because of the reduced distance between adjacent distributions. This is achieved by using incremental step pulse program (ISPP), i.e., memory cells on the same word-line are repeatedly programmed using a program-and-verify approach with a stair case program voltage applied to word-lines. Each programmed state associates with a verify voltage that is used in verify operations and sets the target position of each threshold voltage distribution window.

Read errors can be caused by distorted or overlapped threshold voltage distribution. An ideal memory cell threshold voltage distribution can be significantly distorted or overlapped due to, e.g., program and erase (P/E) cycle, cell-to-cell interference, and data retention errors, which will be discussed in the following, and such read errors may be managed in most situations by using error correction codes (ECC).

FIG. 4 illustrates an example of ideal threshold voltage distribution curves 410 and an example of distorted threshold voltage distribution curves 420. The vertical axis indicates the number of memory cells that has a particular threshold voltage represented on the horizontal axis.

For n-bit multi-level cell NAND flash memory, the threshold voltage of each cell can be programmed to 2^(n) possible values. In an ideal multi-level cell NAND flash memory, each value corresponds to a non-overlapping threshold voltage window.

Flash memory P/E cycling causes damage to a tunnel oxide of floating gate of a charge trapping layer of cell transistors, which results in threshold voltage shift and thus gradually degrades memory device noise margin. As P/E cycles increase, the margin between neighboring distributions of different programmed states decreases and eventually the distributions start overlapping. The data bit stored in a memory cell with a threshold voltage programmed in the overlapping range of the neighboring distributions may be misjudged as a value other than the original targeted value.

FIG. 5 illustrates an example of a cell-to-cell interference in NAND flash memory. The cell-to-cell interference can also cause threshold voltages of flash cells to be distorted. The threshold voltage shift of one memory cell transistor can influence the threshold voltage of its adjacent memory cell transistor through parasitic capacitance-coupling effect between the interfering cell and the victim cell. The amount of the cell-to-cell interference may be affected by NAND flash memory bit-line structure. In the even/odd bit-line structure, memory cells on one word-line are alternatively connected to even and odd bit-lines and even cells are programmed ahead of odd cells in the same word-line. Therefore, even cells and odd cells experience different amount of cell-to-cell interference. Cells in all-bit-line structure suffer less cell-to-cell inference than even cells in the even/odd bit-line structure, and the all-bit-line structure can effectively support high-speed current sensing to improve the memory read and verify speed.

The dotted lines in FIG. 5 denote the nominal distributions of PIE states (before program interference) of the cells under consideration, and the “neighbor state value” denotes the value that the neighboring state has been programmed to. As illustrated in FIG. 5, if the neighboring state is programmed to P1, the threshold voltage distributions of the cells under consideration shift by a specific amount. However, if the neighboring state is programmed to P2, which has a higher threshold voltage than P1, that results in a greater shift compared to the neighboring state being P1. Similarly, the shift in the threshold voltage distributions is greatest when the neighboring state is programmed to P3.

FIG. 6 illustrates an example of a retention error in NAND flash memory by comparing normal threshold-voltage distribution and shifted threshold-voltage distribution. The data stored in NAND flash memories tend to get corrupted over time and this is known as a data retention error. Retention errors are caused by loss of charge stored in the floating gate or charge trap layer of the cell transistor. Due to wear of the floating gate or charge trap layer, memory cells with more program erase cycles are more likely to experience retention errors. In the example of FIG. 6, comparing the top row of voltage distributions (before corruption) and the bottom row of distributions (contaminated by retention error) reveals a shift to the left.

The distortion and interference effects described in FIGS. 4-6 can occur in the DRAM that stores the FTL, and in most cases, a majority of the ECC workload is for correcting any error when the data (e.g., FTL mappings from LBA to PBA) is read from DRAM.

Examples of ECCs that are used in the DRAM include binary linear codes, which are often denoted as (n, k) linear codes, where n is the codeword length and k is the rank or number of rows in the generator matrix of the binary linear code. Some binary codes are quasi-cyclic codes, but there is no known efficient decoder for these quasi-cyclic linear codes.

In some embodiments, the standard decoding method used is syndrome decoding, which requires the storage of a large table (the standard array table) that stores syndromes, error patterns and the like. Decoding via a standard array is a form of nearest neighbor decoding (also referred to as minimum distance decoding). Embodiments of the disclosed technology include efficient techniques to reduce the size of the standard array table, which advantageously reduce the implementation complexity of the quasi-cyclic linear code decoder in hardware, but do not significantly impact the latency of the decoding operations.

Quasi-cyclic codes are defined by the property that, for an integer no, every cyclic shift of a codeword by no places is also a codeword. For a systematic (n=mn₀, k=mk₀) quasi-cyclic linear code, the corresponding parity check matrix H is represented as:

$H = \begin{bmatrix} \; & C_{1,1}^{\prime} & C_{1,2}^{\prime} & \text{…} & C_{1,k_{0}}^{\prime} \\ \; & C_{2,1}^{\prime} & C_{2,2}^{\prime} & \text{…} & C_{2,k_{0}}^{\prime} \\ I_{n - k} & C_{3,1}^{\prime} & C_{3,2}^{\prime} & \; & C_{3,k_{0}}^{\prime} \\ \; & \vdots & \vdots & \; & \vdots \\ \; & C_{{n_{0} - k_{0}},1}^{\prime} & C_{{n_{0} - k_{0}},2}^{\prime} & \ldots & C_{{n_{0} - k_{0}},k_{0}}^{\prime} \end{bmatrix}$

Herein, I_(n−k) represents an (n−k)× (n−k) identity matrix, and each C′_(i,j) is a m×m circulant matrix of the following form:

$C_{i,j}^{\prime} = {\begin{bmatrix} c_{0} & c_{1} & c_{2} & \ldots & c_{m - 1} \\ c_{m - 1} & c_{0} & c_{1} & \ldots & c_{m - 2} \\ c_{m - 1} & c_{m - 1} & c_{0} & \ldots & c_{m - 3} \\ \vdots & \vdots & \vdots & \; & \vdots \\ c_{1} & c_{2} & c_{3} & \ldots & c_{0} \end{bmatrix}.}$

In some embodiments, a syndrome decoder for a quasi-cyclic linear code uses a standard array table, an example of which is illustrated in Table 1. A coset leader (e.g., e_(i) in Table 1) represents the most likely error pattern corresponding to a syndrome (e.g., s_(i) in Table 1) that is computed based on the received sequence and the parity check matrix.

TABLE 1 Example of a Standard Array Table Syndromes Coset leaders Codeword s₀ e₀ = (0)_(n) c₀ c₁ c₂ c₃ . . . c₂k s₁ e₁ c₀ + e₁ c₁ + e₁ c₂ + e₁ c₃ + e₁ . . . c₂k + e₁ s₂ e₂ c₀ + e₂ c₁ + e₂ c₂ + e₂ c₃ + e₂ . . . c₂k + e₂ . . . . . . . . . . . . . . . . . . . . . s₂ _(n−k) e₂ _(n−k) c₀ + e₂ _(n−k) c₁ + e₂ _(n−k) c₂ + e₂ _(n−k) c₃ + e₂ _(n−k) . . . c₂k + e₂ _(n−k)

An example decoding operation in a syndrome decoder includes first finding the received sequence r in the standard array table, and then designating the uppermost element in the column corresponding to r as the correct codeword. As previously discussed, storing the entire standard array table typically results in a large memory requirement.

In some embodiments, only the first two columns of Table 1 (the syndrome and the coset leader) are stored instead of storing the whole standard array table. In contrast to finding r in the standard array table and selecting the uppermost element in the column corresponding to r, the following decoding operations may be used to determine the correct codeword:

1. Compute the syndrome s′=H·r,

2. Find the corresponding coset leader e′, and

3. Generate the most likely transmitted codeword to be r⊕e′, where ⊕ denotes the bit-wise addition operation.

As discussed previously, for quasi-cyclic codes, there exists a cyclic shift property between the syndrome and the corresponding coset leader. Thus, for each syndrome and its corresponding coset leader, denoted (s_(i), e_(i)), there exist m−1 other pairs of syndrome and coset leaders, denoted (s_(j), e_(j)), that are cyclic shifts of (s_(i), e_(i)). In some embodiments, this property is exploited to store only one of the shifted pairs, thereby reducing the size of the standard array table by a factor of m. Since the cyclic shifts are by a predetermined integer (e.g., n₀ in the previously discussed example), storing only a single pair does not result in any ambiguity or lack of information, and the correct syndrome and coset leader can be determined.

An example of a syndrome-coset leader pair table that can be stored in memory to enable a lower complexity implementation of a syndrome decoder is shown in Table 2.

TABLE 2 Example of a syndrome-coset leader pair table Syndromes Coset leaders s₀ e₀ = (0)_(n) s₁ e₁ s₂ e₂ . . . . . . S₂ _(n-k) e₂ _(n-k)

Storing only the syndrome-coset leader pairs instead of the entire standard array table is based on the following property for the error pattern e_(i):

$s_{i} = {\sum\limits_{{j:e_{ij}} = 1}{h_{j}.}}$

Herein, h_(j) is the j-th column of the parity check matrix H, and the syndrome s_(i) is the sum of those columns of the parity check matrix which correspond to a “1” in the error pattern. The validity of the above property relies on the columns of the parity check matrix also having the cyclic shift property. Thus, the size of the standard array table can be reduced to 1/m of the original size by storing only the unique syndrome-coset leader (equivalently, the syndrome-error) pairs instead of all the syndrome-error pairs.

In some embodiments, the size of the standard array table can be further reduced by storing only the coset leader e_(i) in a memory cell instead of the (s_(i), e_(i)) pair. This table reduction is based on using the syndrome s_(i) to compute the address of the memory cell in which the corresponding coset leader e_(i) is stored. In an example, the address of the memory cell (denoted A_(i)) can be computed as a function of the syndrome, e.g., A_(i)=f(s_(i)). Storing only the coset leader instead of the pair reduces the size of the standard array table by a factor of 2.

In some embodiments, both the techniques described may be used to advantageously reduce the size of the original standard array table by a factor of 2m. However, in this case, upon storing just the unique syndrome-error pairs, the syndromes are no longer consecutive and cannot be directly used as an address, and as discussed above, a function of the syndrome is used as an address. In an example, the function is a hash function and an open addressing technique may be implemented to handle any collisions that might arise.

FIG. 7 is a block diagram of an example quasi-cyclic code syndrome-based decoder. As illustrated in FIG. 7, an n-bit received noisy codeword (denoted In[0, n−1] in FIG. 7) is used to compute a syndrome (S) by the syndrome generator 710. For example, if the parity check matrix is denoted as H, then the syndrome is computed as S=H·(In[0,n−1])^(T).

In some embodiments, and because the table has been compressed by exploiting the cyclic shift property of the quasi-cyclic linear code, the syndrome (S) is one of multiple syndromes that are computed. Each of the multiple computed syndromes are cyclic shifts of each other. For an example, the multiple syndromes may be computed in parallel. The computation of the multiple syndromes may be based on the decimal representation of the syndromes, and the syndrome that is selected (denoted Syn[0,n−k] in FIG. 7) is the one with the minimum magnitude in the decimal representation.

The (n−k+1)-bit syndrome (Syn[0,n−k]) is input to an address generator 720, which generates an address and a shift number. In some embodiments, the output address is generated based on a hash function, and the shift number calculated is the maximum possible shift. In other embodiments, the hash function performs a modulo operation (with the memory size as the modulus) to generate the address. In an example, the hash function may be the MD-5 or SHA-3 hash function. In another example, the hash function may be implemented using a linear-feedback shift register (LFSR).

In this example, the compressed table 730 implements both table reduction methods described above, thereby resulting in a table that is 1/(2m) the size of the original standard array table. Since all the coset leaders (correspondingly, their syndromes) are not stored in the compressed table, the cyclic shift is used in both the address generator 720 and the error location computation unit 740 to find the correct address and the correct error pattern, respectively.

The correct error pattern coset leader, the shift number (generated by the address generator 720 and based on the cyclic shift) and the received noisy codeword (In[0,n−1]) are input to the error location computer 740 that computes the correct error pattern. In an example, the correct error pattern is determined based on the correct error pattern coset leader and the shift number. The correct error pattern and the received noisy codeword are used to output the candidate (or most likely) transmitted codeword (denoted Out[0,k−1] in FIG. 7).

FIG. 8 illustrates a flowchart of a method 800 for reducing the complexity of a decoder (e.g., the decoder 700 illustrated in FIG. 7). The method 800 includes, at operation 810, receiving a noisy codeword that is a based on a transmitted codeword generated from a quasi-cyclic linear code.

The method 800 includes, at operation 820, computing a plurality of syndromes based on the noisy codeword.

The method 800 includes, at operation 830, selecting a first syndrome from the plurality of syndromes.

The method 800 includes, at operation 840, generating a memory cell address as a function of the first syndrome.

The method 800 includes, at operation 850, reading, based on the memory cell address, a coset leader corresponding to the first syndrome.

The method 800 includes, at operation 860, determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.

In some embodiments, the plurality of syndromes are computed in parallel.

In some embodiments, the first syndrome is selected based on decimal representations of the plurality of syndromes, and has a minimum decimal representation associated with a shift.

In some embodiments, the method 800 further includes the operations of determining, based on the coset leader, an error pattern; and applying the shift to error pattern to generate a shifted error pattern that is used to determine the candidate version of the transmitted codeword.

In some embodiments, generating the memory cell address is based on a hash function, and collisions of the hash function are mitigated based on an open addressing technique.

In some embodiments, the quasi-cyclic linear code is a quasi-cyclic (256, 240) linear code that is used in a flash translation layer (FTL), and the codeword is a mapping from a logical block address (LBA) to a physical block address (PBA).

In some embodiments, the memory cell address corresponds to a memory that stores one or more coset leaders and excludes syndromes.

In some embodiments, each of the one or more coset leaders corresponds to a plurality of syndromes that are cyclic shifts of a syndrome paired with the each of the one or more coset leaders.

Implementations of the subject matter and the functional operations described in this patent document can be implemented in various systems, digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Implementations of the subject matter described in this specification can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a tangible and non-transitory computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them. The term “data processing unit” or “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.

A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.

The processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).

Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of nonvolatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

While this patent document contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this patent document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. Moreover, the separation of various system components in the embodiments described in this patent document should not be understood as requiring such separation in all embodiments.

Only a few implementations and examples are described and other implementations, enhancements and variations can be made based on what is described and illustrated in this patent document. 

What is claimed is:
 1. A method for reducing complexity of a decoder, comprising: receiving a noisy codeword that is based on a transmitted codeword generated from a quasi-cyclic linear code; computing a plurality of syndromes based on the noisy codeword; selecting a first syndrome from the plurality of syndromes; generating a memory cell address as a function of the first syndrome, the memory cell address corresponding to a memory cell that stores one or more coset leaders and excludes syndromes; reading, based on the memory cell address, a coset leader stored in the memory cell that is accessed using the memory cell address, the coset leader corresponding to the first syndrome; and determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.
 2. The method of claim 1, wherein the plurality of syndromes are computed in parallel.
 3. The method of claim 1, wherein the first syndrome is selected based on decimal representations of the plurality of syndromes, and wherein the first syndrome has a minimum decimal representation associated with a shift.
 4. The method of claim 3, further comprising: determining, based on the coset leader, an error pattern; and applying the shift to the error pattern to generate a shifted error pattern that is used to determine the candidate version of the transmitted codeword.
 5. The method of claim 1, wherein generating the memory cell address is based on a hash function, and wherein collisions of the hash function are mitigated based on an open addressing technique.
 6. The method of claim 1, wherein the quasi-cyclic linear code is a quasi-cyclic (256, 240) linear code that is used in a flash translation layer (FTL), and wherein the transmitted codeword is a mapping from a logical block address (LBA) to a physical block address (PBA).
 7. The method of claim 1, wherein each of the one or more coset leaders corresponds to a plurality of syndromes that are cyclic shifts of a syndrome paired with the each of the one or more coset leaders.
 8. An apparatus for reducing complexity of a decoder, the apparatus comprising: at least one processor; and a non-transitory memory with instructions stored thereon, the instructions upon execution by the at least one processor, causing the at least one processor to: receive a noisy codeword that is based on a transmitted codeword generated from a quasi-cyclic linear code; compute a plurality of syndromes based on the noisy codeword; select a first syndrome from the plurality of syndromes; generate a memory cell address as a function of the first syndrome, the memory cell address corresponding to a memory cell that stores one or more coset leaders and excludes syndromes; read, based on the memory cell address, a coset leader stored in the memory cell that is accessed using the memory cell address, the coset leader corresponding to the first syndrome; and determine, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.
 9. The apparatus of claim 8, wherein the plurality of syndromes are computed in parallel.
 10. The apparatus of claim 8, wherein the first syndrome is selected based on decimal representations of the plurality of syndromes, and wherein the first syndrome has a minimum decimal representation associated with a shift.
 11. The apparatus of claim 10, wherein the instructions upon execution by the at least one processor further cause the at least one processor to: determine, based on the coset leader, an error pattern; and apply the shift to the error pattern to generate a shifted error pattern that is used to determine the candidate version of the transmitted codeword.
 12. The apparatus of claim 8, wherein generating the memory cell address is based on a hash function, and wherein collisions of the hash function are mitigated based on an open addressing technique.
 13. The apparatus of claim 12, wherein an implementation of the hash function is based on a linear-feedback shift register (LFSR).
 14. The apparatus of claim 8, wherein each of the one or more coset leaders corresponds to a plurality of syndromes that are cyclic shifts of a syndrome paired with the each of the one or more coset leaders.
 15. A non-transitory computer readable storage medium having instructions stored thereupon, the instructions, when executed by a processor, causing the processor to implement a method for reducing complexity of a decoder, comprising: instructions for receiving a noisy codeword that is based on a transmitted codeword generated from a quasi-cyclic linear code; instructions for computing a plurality of syndromes based on the noisy codeword; instructions for selecting a first syndrome from the plurality of syndromes; instructions for generating a memory cell address as a function of the first syndrome, the memory cell address corresponding to a memory cell that stores one or more coset leaders and excludes syndromes; instructions for reading, based on the memory cell address, a coset leader stored in the memory cell that is accessed using the memory cell address, the coset leader corresponding to the first syndrome; and instructions for determining, based on the noisy codeword and the coset leader, a candidate version of the transmitted codeword.
 16. The non-transitory computer readable storage medium of claim 15, wherein generating the memory cell address is based on a hash function, and wherein collisions of the hash function are mitigated based on an open addressing technique.
 17. The non-transitory computer readable storage medium of claim 15, wherein each of the one or more coset leaders corresponds to a plurality of syndromes that are cyclic shifts of a syndrome paired with the each of the one or more coset leaders. 